Rectangular waveguides are important for two reasons. First of all, the
Laplacian operator separates nicely in Cartesian coordinates, so that
the boundary value problem that must be solved is both familiar and
straightforward. Second, they are extremely common in actual
application in physics laboratories for piping e.g. microwaves around as
experimental probes.

In Cartesian coordinates, the wave equation becomes:

(10.72)

This wave equation separates and solutions are products of sin, cos or
exponential functions in each variable separately. To determine which
combination to use it suffices to look at the BC's being satisfied. For
TM waves, one solves for subject to , which is
automatically true if:

(10.73)

where and are the dimensions of the and sides of the
boundary rectangle and where in principle
.

However, the wavenumber of any given mode (given the frequency) is
determined from:

(10.74)

where for a ``wave'' to exist to propagate at all. If either
index or is zero, there is no wave, so the first mode that can
propagate has a dispersion relation of:

(10.75)

so that:

(10.76)

Each combination of permitted and is associated with a cutoff of
this sort - waves with frequencies greater than or equal to the cutoff
can support propogation in all the modes with lower cutoff frequencies.

If we repeat the argument above for TE waves (as is done in Jackson,
which is why I did TM here so you could see them both) you will be led
by nearly identical arguments to the conclusion that the lowest
frequency mode cutoff occurs for , and to produce
the
solution to the wave equation above. The
cutoff in this case is:

(10.77)

There exists, therefore, a range of frequencies in between where only one TE mode is supported with dispersion:

(10.78)

Note well that this mode and cutoff corresponds to exactly one-half a
free-space wavelength across the long dimension of the waveguide. The
wave solution for the right-propagating TE mode is:

(10.79)

(10.80)

(10.81)

We used
and
to get the second of
these, and
) to get the last one.

There is a lot more one can study in Jackson associated with waveguides,
but we must move on at this time to a brief look at resonant cavities
(another important topic) and multipoles.